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Problem of the Day #400: Squirrel City April 22, 2012

Posted by Saketh in : potd , add a comment

In Squirrel City there is a $100$ by $100$ grid on which every treehouse is located, one at each lattice point. The administration is doing some construction and needs to chop down trees for wood.

They know that no citizen will complain unless there exists a treehouse from which at least two different stumps can be observed. Furthermore, there must be at least one treehouse left for the squirrels to use.

Determine the greatest number of trees which can be chopped down without inciting any complaints. Assume that the radius of each tree is negligible when compared to the gaps between the trees.

Problem of the Day #393: A Number Game April 15, 2012

Posted by Saketh in : potd , add a comment

Scott is playing with numbers. To begin, he selects any four-digit number (leading zeros are allowed).

Each turn, he sorts the digits of his current number in both ascending and descending order to produce two other values. He then subtracts the smaller of these from the larger, and keeps the result as his new number.

Eventually, Scott’s routine reaches the fixed point $6174$. We call it a fixed point because $7641 – 1467$ simply produces $6174$ again. How many possible values are there for his starting number?

Problem of the Day #388: Find the Biggest Prime April 10, 2012

Posted by Saketh in : potd , add a comment

For certain values of $a$ and $b$, the expression

$$\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$$

produces an integer. If we consider the set of all such integers, what is the largest prime member?

Problem of the Day #385: A Variation on Fermat’s Last Theorem April 7, 2012

Posted by Saketh in : potd , 1 comment so far

Alex has been tinkering with an elegant proof of Fermat’s Last Theorem. Although this post is a little too small to contain his proof, he does have for you an alternate equation to consider:

$$ x^x + y^y = 2 \cdot z^z $$

Find, with proof, all ordered triples of positive integers $(x,y,z)$ that satisfy this equation.

Problem of the Day #381: Number of Integer Solutions April 3, 2012

Posted by Saketh in : potd , add a comment

Determine the number of ordered pairs of integers $(x,y)$ such that

$${(x^2-y^2)}^2 = 1 + 25y $$

Problem of the Day #379: Easy Problem April 1, 2012

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Dear Loyal Viewers,

It’s been more than a year since we started “Math Problem of the Day” and it has been an amazing experience for all of us. However, we regret to inform you that we will be stopping our daily “problem of the day” in search of other pursuits.

For our last problem of the day, please convert $137,665,927,309,005,970,969,842,064$ to base $36$.

Problem of the Day #376: Log Entry 01 March 29, 2012

Posted by Albert in : potd , add a comment

Alex has decided to provide you with the following data:

$$
\begin{eqnarray}
\ln(2) & \approx & 0.693147181 \\
\ln(3) & \approx & 1.09861229 \\
\ln(5) & \approx & 1.60943791 \\
\ln(7) & \approx & 1.94591015
\end{eqnarray}
$$

Your mission, should you choose to accept it, is to find the number of digits in the decimal expansion of $35^{12000}$.

Problem of the Day #375: Log Entry 00 March 28, 2012

Posted by Albert in : potd , add a comment

Alex has decided to provide you with the following data:

$$
\begin{eqnarray}
\ln(2) & \approx & 0.693147181 \\
\ln(3) & \approx & 1.09861229 \\
\ln(5) & \approx & 1.60943791 \\
\ln(7) & \approx & 1.94591015
\end{eqnarray}
$$

Your mission, should you choose to accept it, is to find the smallest positive integer $x$ such that $2^{10 x}$ has more than $3x+1$ digits.

Problem of the Day #364: Divisibility Pairs March 17, 2012

Posted by Saketh in : potd , add a comment

Suppose that Albert writes down all numbers of the form $p^0+q^0, p^1+q^1, p^2+q^2, \ldots$ up to $p^n+q^n$, where $p$ and $q$ are distinct primes. Determine, in terms of $n$, the number of unordered pairs of distinct members of this sequence such that one divides the other.

Problem of the Day #362: Digitally Odd Numbers March 15, 2012

Posted by Saketh in : potd , add a comment

Suppose that $n$ is a five digit multiple of $5$, all of whose digits are odd. Suppose further than $\frac{n}{5}$ also has five odd digits. How many possible values of $n$ are there?